Standard spectral methods exhibit exponential convergence for analytic problems but stall when approximating singular solutions to parametric PDEs. We quantify this classical approximation barrier, demonstrating a $60{,}806\times$ error reduction when the basis is augmented with the correct singular atom (oracle upper bound). In an idealized exponent-scanning surrogate, the recovered exponent achieves mean absolute error $0.006$ under moderate noise, and our sensitivity study shows that such errors still yield large gains when used to augment the basis. We present numerical evidence that, in our specific power-law dictionary, $\ell_1$-minimization (LASSO) successfully recovers singular terms despite near-perfect dictionary coherence ($\mu \approx 0.997$), with a sharp empirical phase transition at small sample sizes, overcoming the limitations of greedy methods. Numerical validation extends to two-dimensional Poisson problems with corner singularities. \end{abstract}